Surface area is the TOTAL AREA

of all the faces (surfaces) of a 3D shape

Measured in SQUARE units (cm², m², ft², etc.)

SA = 6s²

Where:

s = side length

A cube has 6 identical square faces

Example: Find surface area of a cube with side 4 cm.

SA = 2(lw + wh + lh)

Where:

l = length

w = width

h = height

SA = bh + (s₁ + s₂ + s₃) × H

Where:

b = base of triangle, h = height of triangle

s₁, s₂, s₃ = sides of triangular base

H = height (length) of prism

SA = B + ½Pl

Where:

B = area of base

P = perimeter of base

l = slant height

SA = s² + 2sl

Where:

s = side of square base

l = slant height

Example: Square pyramid with base 6 cm and slant height 8 cm.

Lateral area is the area of the SIDES ONLY

(NOT including the base or bases)

Also called Curved Surface Area or LSA

LA = Ph

Where:

P = perimeter of base

h = height of prism

LA = ½Pl

Where:

P = perimeter of base

l = slant height

Relationship:

Total Surface Area = Lateral Area + Base Area(s)

SA = 2πr² + 2πrh

or

SA = 2πr(r + h)

Where:

r = radius of circular base

h = height of cylinder

Breaking it down:

• 2πr² = area of both circular bases (top and bottom)

• 2πrh = lateral (curved) surface area

LSA = 2πrh

Only the curved surface (no bases)

Example: Cylinder with radius 3 cm and height 10 cm.

Volume is the amount of SPACE INSIDE

a 3D shape (capacity)

Measured in CUBIC units (cm³, m³, ft³, etc.)

V = s³

or V = s × s × s

Where s = side length

V = l × w × h

Where:

l = length, w = width, h = height

V = B × h

Where:

B = area of base

h = height (perpendicular to base)

Example: Rectangular prism 8 cm × 5 cm × 6 cm.

V = (1/3) × B × h

Where:

B = area of base

h = height (perpendicular from base to apex)

Key Point:

Volume of pyramid = (1/3) × Volume of prism

with the same base and height!

Example: Square pyramid with base 6 cm and height 9 cm.

V = πr²h

Where:

r = radius of circular base

h = height of cylinder

π ≈ 3.14

Think of it as:

V = (Area of circular base) × height

V = πr² × h

Example: Cylinder with radius 4 cm and height 10 cm.

Surface Area Formulas

Volume Formulas

✓ Surface Area: Total area of ALL faces (square units: cm², m²)

✓ Volume: Space inside (cubic units: cm³, m³)

✓ Lateral Area: Only side faces (no bases)

✓ Prism volume: V = Bh (base area × height)

✓ Pyramid volume: V = (1/3)Bh (one-third of prism)

✓ Cylinder: Think of it as a circular prism

✓ Height: Always perpendicular to the base

✓ Slant height: Along the slanted surface (pyramids)

✓ Base: Can be any polygon (triangle, square, etc.)

✓ Units: Always check and include proper units!

Surface Area vs Volume:

"Surface is what you paint, Volume is what you fill - remember this skill!"

Cube Formulas:

"Six faces squared for surface true, side cubed for volume - easy to do!"

Prism vs Pyramid Volume:

"Prism is full, pyramid's one-third - that's the volume word!"

Cylinder Surface Area:

"Two pi r for both bases round, two pi r h for curved part found, add them up - that's profound!"

Lateral Area:

"Lateral means sides alone, bases not in this zone!"

Base × Height Pattern:

"Prism full, pyramid third - Bh patterns you've heard!"